Gradient method of optimal control applied to the operation of a dam water gate

Abstract

An extension of the authors' previous methods is presented for the optimal control of flood propagation via a dam gate, based on a combination of the finite element and gradient methods. It is assumed in previous papers that the control duration is the same as the duration of the flood. However, the duration of the control does not necessarily coincide with that of the flood flow. To overcome this difficulty, the gradient method is applied to solve the free terminal time-fixed terminal condition problem. It is shown that the water elevation can be controlled exactly the same as with the previously presented method. It is also shown that the computation can be terminated at a far shorter time than the terminal time of the flood. KEY WORDS

Vertical slot fishways: Mathematical modeling and optimal management

“…In order to minimize the objective function J the standard proposal consists of using a gradient-type algorithm, where the gradient ∇J (q) can be directly obtained from expression (12) via the computation of the adjoint system (6)-(7). However, due to the high computational cost arisen from the numerical resolution of this adjoint system, in this paper we alternatively propose a gradient-free algorithm for solving the discretized optimization problem, where the adjoint system is not needed to be solved.…”

“…In order to develop our mathematical study we make use of several tools related to the optimal control theory and the optimization techniques, which have been very useful in the mathematical resolution of other environmental problems previously addressed by the authors [18,3,2]. Other nice applications of optimal control theory to environmental problems can be found, among others, in [8,12,22]. Section 2 is devoted to present a mathematical formulation of the optimal control problem for a standard nine pools channel, where the state system is given by the shallow water equations determining the height of water and its velocity (averaged in height), the control is the normal flux of water on the inflow, and the objective function is related to the existence of rest areas for fish and a water velocity suitable for fish leaping and swimming capabilities.…”

Optimal control applied to water flow using second order adjoint method

“…There are two types of solution method applied in the control problems, the sensitivity coefficient method and the adjoint equation method (Shimada et al 1991, Wang et al 1992, Kawahara and Shimada 1994a, 1994b, Kawahara et al 1995, Wang et al 1995, Katopodes and Piasecki 1996, Kawahara and Sasaki 1996, Hirano et al 1996, Kojima et al 1996, Suzuki et al 1996, Anju and Kawahara 1997, Piasecki and Katopodes 1997a, 1997b, Sakuma and Kawahara 1997, Suzuki and Kawahara 1997, Maruoka et al 1998, Miyata et al 1999, Ishii and Kawahara 2001, Kato et al 2001, Alelseev and Navon 2002, Le Dimet et al 2002, Kurahashi and Kawahara 2003, Fujii and Kawahara 2004. The sensitivity coefficient method is usually used for the conventional control problem.…”